A Necessary Condition for the Completeness of the System $\{e^{-\lambda_nt}\mid\operatorname{Re}\lambda_n>0\}$ in the Spaces~$C_0(\mathbb R_+)$ and $L^p(\mathbb R_+)$, $p>2$

I. O. Krasnobaev

Geodesic

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A Necessary Condition for the Completeness of the System $\{e^{-\lambda_nt}\mid\operatorname{Re}\lambda_n>0\}$ in the Spaces~$C_0(\mathbb R_+)$ and $L^p(\mathbb R_+)$, $p>2$

I. O. Krasnobaev

Matematičeskie zametki, Tome 83 (2008) no. 6, pp. 831-842

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Résumé

We obtain a necessary condition for the completeness of the system $$ e(\Lambda)=\{e^{-\lambda_nt}\mid\operatorname{Re}\lambda_n>0,\,n\in\mathbb Z\} $$ in the spaces $C_0$ and $L^p(\mathbb R_+)$, $p>2$, for the case in which the set of limit points of the sequence $\{\lambda_n\}$ is countable and separable.

Keywords: sequence of exponentials, the spaces $C_0(\mathbb R_+)$ and $L^p(\mathbb R_+)$, $p>2$, Hardy class of functions, Bernstein's inequality, analytic function.
Mots-clés : Szász condition

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     author = {I. O. Krasnobaev},
     title = {A {Necessary} {Condition} for the {Completeness} of the {System} $\{e^{-\lambda_nt}\mid\operatorname{Re}\lambda_n>0\}$ in the {Spaces~}$C_0(\mathbb R_+)$ and $L^p(\mathbb R_+)$, $p>2$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {831--842},
     publisher = {mathdoc},
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     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2008_83_6_a3/}
}

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I. O. Krasnobaev. A Necessary Condition for the Completeness of the System $\{e^{-\lambda_nt}\mid\operatorname{Re}\lambda_n>0\}$ in the Spaces~$C_0(\mathbb R_+)$ and $L^p(\mathbb R_+)$, $p>2$. Matematičeskie zametki, Tome 83 (2008) no. 6, pp. 831-842. http://geodesic.mathdoc.fr/item/MZM_2008_83_6_a3/